Question: $\dfrac{ -4q + 5r }{ -7 } = \dfrac{ 4q + 3s }{ 9 }$ Solve for $q$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -4q + 5r }{ -{7} } = \dfrac{ 4q + 3s }{ 9 }$ $-{7} \cdot \dfrac{ -4q + 5r }{ -{7} } = -{7} \cdot \dfrac{ 4q + 3s }{ 9 }$ $-4q + 5r = -{7} \cdot \dfrac { 4q + 3s }{ 9 }$ Multiply both sides by the right denominator. $-4q + 5r = -7 \cdot \dfrac{ 4q + 3s }{ {9} }$ ${9} \cdot \left( -4q + 5r \right) = {9} \cdot -7 \cdot \dfrac{ 4q + 3s }{ {9} }$ ${9} \cdot \left( -4q + 5r \right) = -7 \cdot \left( 4q + 3s \right)$ Distribute both sides ${9} \cdot \left( -4q + 5r \right) = -{7} \cdot \left( 4q + 3s \right)$ $-{36}q + {45}r = -{28}q - {21}s$ Combine $q$ terms on the left. $-{36q} + 45r = -{28q} - 21s$ $-{8q} + 45r = -21s$ Move the $r$ term to the right. $-8q + {45r} = -21s$ $-8q = -21s - {45r}$ Isolate $q$ by dividing both sides by its coefficient. $-{8}q = -21s - 45r$ $q = \dfrac{ -21s - 45r }{ -{8} }$ Swap signs so the denominator isn't negative. $q = \dfrac{ {21}s + {45}r }{ {8} }$